SIMPLIFY EXPRESSIONS INVOLVING EXPONENTIAL FUNCTION

Some more examples.

Simplify :

Example 1 :

(4^m + 8^m)/4^m

Solution :

=  (4^m + (4⋅2)^m)/4^m

=  (4^m + 4^m⋅2)^m)/4^m

=  [4^m(1 + 2^m)]/4^m

=  1 + 2^m

So, the answer is 1 + 2^m

Example 2 :

(4^m + 8^m)/(1 + 2^m)

Solution :

=  4^m+(4⋅2)^m/(1 + 2^m)

=  (4^m+4^m⋅2^m)/(1 + 2^m)

=  4^m(1 + 2^m)/(1 + 2^m)

=  4^m

So, the answer is 4^m

Example 3 :

(7^b + 21^b)/7^b

Solution :

Given, (7^b + 21^b)/7^b

=  (7^b + (7⋅3)^b)/7^b

=  (7^b + 7^b⋅3^b)/7^b

Factoring 7^b from the numerator, we get

=  [7^b(1 + 3^b)]/7^b

=  1 + 3^b

So, the answer is 1 + 3^b

Example 4 :

(4ⁿ⁺² – 4ⁿ)/4ⁿ

Solution :

Given, (4ⁿ⁺² – 4ⁿ)/4ⁿ

By using exponent of the product rule, we get

=  [(4ⁿ. 4²) – 4ⁿ]/4ⁿ

=  [(4ⁿ . 16) – 4ⁿ]/4ⁿ

=  [4ⁿ(16 – 1)]/4ⁿ

=  15

So, the answer is 15

Example 5 :

(4ⁿ⁺² – 4ⁿ)/15

Solution :

Given, (4ⁿ⁺² – 4ⁿ)/15

By using exponent of the product rule, we get

=  [(4ⁿ. 4²) – 4ⁿ]/15

Factoring 4ⁿ from the numerator, we get

=  [(4ⁿ . 16) – 4ⁿ]/15

=  [4ⁿ(16 – 1)]/15

=  4ⁿ(15)/15

=  4ⁿ

So, the answer is 4ⁿ

Example 6 :

(2^m+n – 2ⁿ)/2ⁿ

Solution :

Given, (2^m+n – 2ⁿ)/2ⁿ

By using exponent of the product rule, we get

=  [(2^m. 2ⁿ) – 2ⁿ]/2ⁿ

Factoring 2ⁿ from the numerator, we get

=  [2ⁿ(2^m – 1)]/2ⁿ

=  2^m - 1

So, the answer is 2^m – 1

Example 7 :

(3ⁿ⁺² – 3ⁿ)/3ⁿ⁺¹

Solution :

Given, (3ⁿ⁺² – 3ⁿ)/3ⁿ⁺¹

By using exponent of the product rule, we get

=  [(3ⁿ. 3²) – 3ⁿ]/(3ⁿ . 3¹)

=  [(3ⁿ . 9) – 3ⁿ]/(3ⁿ . 3)

=  [3ⁿ(9 – 1)]/[3ⁿ(3)]

=  8/3

So, the answer is 8/3

Example 8 :

If (-a² b³)(2ab²)(-3b) = ka^m bⁿ, what is the value of m + n ?

Solution :

(-a² b³)(2ab²)(-3b) = ka^m bⁿ

6a² ab³b² b = ka^m bⁿ

6a²⁺¹ b³⁺²⁺¹ = ka^m bⁿ

6a³ b⁶ = ka^m bⁿ

Comparing the corresponding terms, we get

k = 6, m = 3 and n = 6

m + n = 3 + 6

= 9

So, the value of m + n is 9.

Example 8 :

If ((2/3)a² b)² (4/3)(ab)^-3 = ka^m bⁿ, what is the value of k ?

Solution :

((2/3)a² b)² (4/3)(ab)^-3 = ka^m bⁿ

(4/9)a⁴ b² (4/3) [1/(ab)³] = ka^m bⁿ

(16/27)a⁴ b² [1/a³b³] = ka^m bⁿ

(16/27) a/b = ka^m bⁿ

(16/27) a¹ b^-1 = ka^m bⁿ

Comparing the corresponding terms, we get

k = 16/27

Example 9 :

If [(x³) (-y)² z^-2] / (x)^-2 (y)³ z = x^m /yⁿ z^p what is the value of m + n + p ?

Solution :

[(x³) (-y)² z^-2] / (x)^-2 (y)³ z = x^m /yⁿ z^p

x³ x²  y² y³z^-2z^-1 = x^m /yⁿ z^p

Using the rules of exponents, we get

x³⁺²  y²⁺³ z^-2-1 = x^m y^-n z^-p

x⁵  y⁵z^-3 = x^m y^-n z^-p

By comparing the corresponding terms, we get

m = 5, -n = 5, -p = -3

m = 5, n = -5 and p = 3

m + n + p = 5 - 5 + 3

= 3

So, the value of m + n + p is 3.

Example 10 :

If 2^x = 5, what is the value of 2^x + 2^2x + 2^3x ?

Solution :

Given that 2^x = 5

= 2^x + 2^2x + 2^3x

= 2^x + (2^x)² + (2^x)³

Applying the value 2^x = 5, we get

= 5 + 5² + 5³

= 5 + 25 + 125

= 155

So, the value of expression is 155.

Example 11 :

(3^x + 3^x + 3^x) 3^x

Which of the following is equivalent to the expressions shown above ?

a)  3^4x      b) 3^{3x^2}     c) 3^{1 + 3x}       d) 3^{1 + 2x}

Solution :

= (3^x + 3^x + 3^x) 3^x

Combining the like terms, we get

= [3 (3^x)] 3^x

Considering the bases, they are the same.

= (3^1+x)3^x

= 3^{1 + x + x}

= 3^{1 + 2x}

So, option d is correct.

Example 12 :

(6xy²)(2xy)² / 8x²y²

If the expression above is written in the form ax^m yⁿ, what is the value of m + n ?

Solution :

= (6xy²)(2xy)² / 8x²y²

= (6xy²)(4x²y²) / 8x²y²

= (24x¹⁺² y²⁺²) / 8x²y²

= (24x³ y⁴) / 8x²y²

= 3xy²

Comparing with ax^m yⁿ

a = 3, m = 1 and n = 2

m + n = 1 + 2

= 3

So, the value of m + n is 3.

Example 13 :

What is the value of 3^{(a - b)} 3^{(a + b)} / 3^{2a + 1}  ?

a)  1/3    b)  1/9     c)  3    d)  9

Solution :

3^{(a - b)} 3^{(a + b)} / 3^{2a + 1}

= 3^{(a - b + a + b)}/ 3^{2a + 1}

= 3^2a/ 3^{2a + 1}

= 3^{2a - (2a + 1)}

= 3^{2a - 2a - 1}

= 3^{- 1}

= 1/3

Example 14 :

What is the value of 2^{(a - 1)(a + 1)}/ 2^{(a - 2)(a + 2)}  ?

a)  1/16    b)  1/8     c)  8    d)  16

Solution :

= 2^{(a - 1)(a + 1)}/ 2^{(a - 2)(a + 2)} 

Multiplying (a - 1)(a + 1), we get

= a² - 1²

Multiplying (a - 2)(a + 2), we get

= a² - 2²

= a² - 4

Since the numerator and denominator has same base, we have to use the same base and subtract the powers.

= (a² - 1²) -  (a² - 4)

= a² - 1² - a² + 4

= 3

= 2³

= 8

So, the answer is option c.

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